The 6-year old had Fall Break last week, so no homework and no enVisionMATH-blogging for me. Tonight, however, she brought home a new worksheet for her weekly homework, and a couple of things caught my eye. I thought I’d throw those out there to you all, along with a question or two, as a two-part blog post.

For the first post, take a look at this (click to enlarge):

Questions:

In your own words, preferably those that a smart 6-year old could understand, what is the basic principle that this page is trying to get across?

What technique does this worksheet want kids to use when doing the Algebra problems?

What’s your opinion about the principle/technique you think the worksheet is trying to communciate? Reasonable? Natural? Likely to be useful, or used frequently later on?

The last post about enVisionMATH and how I, as a math person and dad, go about trying to make sense of what my 6-year old brings home from first grade seems to have struck a chord among parents. The comments have been outstanding and there seems to be a real need for this kind of conversation. So I have a few more such posts coming up soon, starting with this one.

The 6-year old brought this home on Monday. Click to enlarge:

It’s about adding “near doubles”, like 3 + 4 or 2 + 3. In case you can’t read the top part or can’t enlarge the photo, here are the steps — yes, there are steps, and that’s kind of the point of this post — for adding near doubles:

“You can use a double to add a near double.” It gives: 4 + 5 and shows four blue balls and five green balls.

“First double the 4”. It shows 4 + 4 = 8, and the four blue balls, and four of the green balls with the extra green ball sort of falling to the ground.

Then it says: “4 + 5 is 4 + 4 and 1 more.” At this point you really have to look at the worksheet itself, because it’s hard to put into words what is going on:

And from there, in the fourth frame, one of the girls in the earlier frame concludes that 4 + 5 must be 9 because 8 and 1 more is 9.

The Guided Practice section has the kids doing four near-double sums. Clearly, the way the worksheet wants kids to learn how to do this is not simply to add 2 + 3, but (1) to recognize that 3 is 2 plus 1 more, (2) add 2 + 2, and (3) then add 1 to the result of 2 + 2:

There’s a thing at the bottom asking kids to explain the process and then a bunch of near-double sums to practice — presumably kids are supposed to use the method described above, but there’s nothing forcing them to do so — and some “algebra” questions with blanks in the place of variables.

I’m not sure exactly how my brain goes about adding near doubles — whether it just somehow does the addition in ways that are almost automatic thanks to 35 years of repetition, or whether there are little tricks it employs — but I am absolutely certain that I don’t do it the way enVisionMATH is telling kids how to do it. I tried it. When I read the worksheet, I thought about near-doubles that aren’t so easy, like 121 and 122. Quick! Add those together. Did you think, “122 is 121 plus one more; 121 + 121 = 242; 242 and 1 more is 243” ? I didn’t — not by a long shot. I just added the numbers together. No methods, no tricks; just old-school addition. There may be some tricks that my brain invokes to “just add the numbers” — for example, I tend to visualize the two terms of the sum stacked atop each other in the classic vertical arrangement for adding, and then visually add the digits — but I am most definitely not going through the four-step process on this worksheet.

In fact, the four-step process complicates matters so much that it’s inexplicable why they are even bringing it up. Most kids at this stage can add 2 + 3 or 5 + 6 in one step. But by introducing this method, there are four operations: comparison (find the larger of the two near-doubles), subtraction (take 1 from the larger number), addition (add the two duplicates), and another addition (add 1 to the result). Technically there is a fifth operation kids have to perform, namely recognize that the two numbers they are adding are near-doubles in the first place.

One might argue that doubling a number (in the third step) is easier than adding it to itself — kids just recognize that doubling 5 gives 10, for instance — and subtracting 1 is a very easy special case of subtraction in general that nearly everybody at this age can do without thinking, similarly for adding 1 at the end. That may be so, but it can’t be so much easier that adding in steps 1, 2 and 4 results in a net reduction in complexity or a net gain in conceptual understanding.

But what about kids who can’t add two one-digit numbers together in one step? There are some of those out there, including probably a few in my daughter’s class. This method doesn’t help those kids. Again, we may argue that adding 4 + 5 is considerably harder than the combined process of comparison, subtracting 1 from 5, doubling 4, then adding 1. But I don’t think so. A four-step process is no less cognitively demanding than a single-step process, even if the four steps are easy. And besides, life does not throw near-doubles at you to add. How is a kid going to learn to add 2 + 5, or 2/5 + 7/8, or 123.38 and 99.99 this way?

If there is some research that suggests that people really do add near doubles this way, I would love to see it. Otherwise it’s hard for me to believe that any more than a tiny fraction of the human population actually does it this way. Is there going to be some mind-blowingly cool way to do complicated arithmetic in one’s head farther down the road that uses this idea, like multiplying numbers that are near-squares or something? Perhaps I should be more patient. But for the time being, I told the 6-year old just to add the numbers together like she already knows how to do, the old-fashioned way.

It’s been back-to-school time for everybody in our household (hence an excuse for the light posting). We started classes at the college today, and last week the 4.5-year old went back to preschool full-time and the 6.5-year old started first grade. (The 1.5-year old is rocking the local daycare.) One of the biggest changes for the kids is for our first-grader, Lucy, since she has real homework for the first time. It’s not much; the expectation is about 20 minutes a night, Monday through Thursday. Some of that homework is math, which I was very excited about — but then that excitement turned to alert caution when I learned my daughter’s class was using enVisionMATH.

I wrote this post on enVisionMATH almost three years ago, basically laughing it off the blogosphere for its happy-clappy, uncritical acceptance of unproven digital nativist frameworks and for going way over the top with smartboards. Little did I know that my own offspring would be in the middle of it just three years later. So, in an effort to process what she’s doing (for me, for her, and for anybody else who cares), this is the first of what might be many posts about the specifics of enVisionMATH, as viewed by a parent whose kid happens to be learning from that curriculum, and who also happens to be a mathematician and math teacher.

I’ll start with the worksheet Lucy brought home this evening, called “Making 8”:

I’ve never had a kid in first grade before, nor can I remember how I did this stuff in first grade, nor have I recently worked with a kid in first grade. So I’m going to share my thoughts, but realize I have no reference for what’s “normal” pedagogy for 6-year olds and what’s not.

This worksheet is really about subtraction, although it never comes out and says so. The first two exercises are attempting to build a sense about subtraction by getting kids to think about how parts fit together to form a particular quantity. enVisionMATH appears to be really big on getting kids to recognize numbers visually rather than by counting. I’ll need to blog about this in a later post, but Lucy’s had some other exercises that, for example, stress the ability to recognize this:

…as the number 6, just by looking at it and without counting the dots, almost to the point of telling kids that they shouldn’t be counting anything but rather arranging things into patterns. Again, that’s for another post.

So, back to the worksheet, kids are supposed to look at the first collection of balloons and, knowing that there are eight of them, see — and only “see” — that 8 splits into 2 plus 6, and then 4 plus 4. I did a few more of these with Lucy using coins (no balloons on hand, sadly). Biggest challenge here: Keeping Lucy from just counting the black balloons and then counting the white balloons. And the only reason this was a challenge was because, as a math person, I knew what the worksheet was getting at: recognizing quantities through visual patterns rather than counting, so the unwritten rule is for kids not to count the balloons. But other parents probably didn’t know this, and their kids just counted. I don’t think this is necessarily wrong, but it doesn’t necessarily help in the next sections either.

The next section is rather startlingly labelled “Algebra”. Remember: This is a worksheet for a first grade class. Why we are bringing up the word “algebra” at this point is anybody’s guess. I suspect this is more to make parents, school boards, and accreditors happy than it is to start getting kids to feel comfortable with the word “algebra”. But anyway, as you can see, the two problems are just the first two problems in reverse.

Lucy had a hard time with this. First of all, she didn’t understand what “the whole” meant. This is not the first time Lucy’s struggled not with the math but with relatively esoteric vocabulary in her math lessons. Last week she had a worksheet where she was to arrange three integers “in order from greatest to least” and “from least to greatest” and we had to take a moment to figure out what all of that meant. Maybe other people’s kids don’t struggle with that, but on the other hand it’s been verified that Lucy is reading at a third or fourth grade level right now, so I wonder if it’s just her.

We had to work these out using manipulatives. We started with fingers because that’s the first thing I thought of. So, I said, if the whole is 8:

…and one part is 3:

…then what was the other part?

Lucy was able to get the answer of “5” with no problem. But… I don’t think she got it the right way. Because when we moved to the next problem and the “one part” was 1, for her, the other part was still 5! This was because when I held up one finger on my left hand this time, there were of course five fingers on the right hand. I tried holding up eight and wiggling one finger instead of putting the “one part” on one hand, but that just confused her. So, we went back to coins and built a “balloon diagram” like in the first two problems, and she got them just fine (and without counting).

I don’t think exercises 3 and 4 are bad problems necessarily, but I do think they came in here way too early. Perhaps I’m missing the context of the actual classroom interaction between Lucy and her teacher, but it would seem like a better idea to do as many exercises like 1 and 2 as possible before moving on to the “algebra”. After all, if you stick to positive integers, there are only seven ways to fill in the blanks __ + __ = 8. (And doing all seven might help kids discover the commutative property early on, which seems like a much more important thing to bring up than “algebra” in first grade.)

And then, it’s not clear to me that doing “algebra” is a better idea here than just doing straight-up subtraction. What’s to be gained by saying “the whole is 8; one part is 3; the other part is ____” versus “What is 8 minus 3?” Again, maybe I’m out of touch, but subtraction is a fundamental skill that algebra builds upon; doing algebra before subtraction seems a little backwards to say the least. A kid who is comfortable with subtraction will be able to do these whole/part problems in a snap by using subtraction. A kid doing these “algebra” problems basically has to invent subtraction in order to do them, or else draw pictures of balloons and start counting. It feels like the curriculum is trying to be intentionally nontraditional here, just for the sake of doing things differently rather than because it works better.

Then we come to the “Journal” question, which is downright sophisticated: “The whole is 8. One part is 8. What is the other part?” Here we reach serious abstraction: You can’t draw balloons like in exercises 1 and 2, and in fact resorting to physical props is tricky.As Derek Bruff mentioned in a tweet about this earlier this evening, the use of the word “part” in conjunction with the quantity 0 is already sort of questionable. What does it even mean to say the “part” is 0? What “part”? I don’t see a “part”. The natural way of interpreting what a “part” is, is as a bunch of objects. If there are no objects present, then there really isn’t a “part”.

We had to resort to thinking not about objects but containers that hold the objects. I took two books sitting nearby. I took my eight coins and said: The whole is 8. One part is 2 — and put 2 coins on one of the books. What is the other part? — and put the remaining coins on the other book. Lucy got the right answer quickly, and she did so by looking back at exercise 1 with the balloons and noticing it was the same problem with different objects, which I thought was pretty smart. I’ll make an algebraist out of her yet! Then I repeated with one part being 1. Then I did it with one part being 6; then 7. Then I said, “The whole is 8; one part is eight.” — putting all eight coins on one book. “What’s the other part?” — showing her my empty hands and an empty book. “Zero,” she said right away.

For her, and maybe not just her, “zero” represents not a size of a part but a state of emptiness of a container — or perhaps the size of a set. It’s how much you see when nothing is there. To map the “zero” concept onto a concept of “part” that presupposes something is there just doesn’t make sense. If this sounds like the New Math, I think we’re barking up the right tree.

The “Tell how you know” was especially tough because it involves getting Lucy to talk about what she did, even though she’s doing it at a sort of visceral level, and then spell the words she needs to use — which is the other type of homework she has. I got her to say out loud what she was thinking, and then I had her say it back to me and then helped her spell the words.

So we made it through the worksheet, but there are a lot of questions in my mind about the pedagogical design of this stuff. And how in the world does this sort of thing work in a household where the parents don’t have the time, patience, interest, fluency, or comfort level in mathematics to sit down and work all this out with the kid?

The LA Times reports on a study suggesting that female elementary school teachers who are anxious about math transmit that anxiety to the girls in their classes:

Girls have long embraced the stereotype that they’re not supposed to be good at math. It seems they may be getting the idea from a surprising source — their female elementary school teachers.

First- and second-graders whose teachers were anxious about mathematics were more likely to believe that boys are hard-wired for math and that girls are better at reading, a new study has found. What’s more, the girls who bought into that notion scored significantly lower on math tests than their peers who didn’t.

The gap in test scores was not apparent in the fall when the kids were first tested, but emerged after spending a school year in the classrooms of teachers with math anxiety. That detail convinced researchers that the teachers — all of them women — were the culprits.

It’s no surprise that teachers who are weak in or nervous about a subject do not inspire confidence, or performance, in that subject among their students. What’s different here is the gender connection — female teachers having a pronounced effect upon girl students — and the subject area. It would be interesting to see just how many elementary school teachers view themselves as “anxious” about teaching math, and then to see how that self-description breaks down by gender. Do a lot of female elementary teachers feel anxious about math? Is it more than male elementary school teachers? I don’t know, but that is certainly the stereotype.

At any rate, the opposite seems to be implied by this study too — female teachers who are strong with math and comfortable with teaching it to kids will have an enhanced positive effect on girls’ perceptions of math and their performance with it. And it seems like a no-brainer that elementary education curricula ought to stress a strong degree of math content mastery among all preservice teachers — of both genders — and demand a high level of fluency with doing and teaching math.Teaching math to little kids is hard, and you have to know a lot of math outside of what you are going to teach if you’re going to do it well. We need to have done with another stereotype: that you major in elementary education because “you just love kids” (you need more than sentimentality to be a good teacher) or because it’s supposedly an easy major (it isn’t, or at least shouldn’t be).

I’ve been lax in posting lately because I’ve been enjoying an all-too-brief interim period between the end of my summer Calculus class and the beginning of Fall semester at the end of this month. I’ve been splitting time this summer between being a stay-at-home dad to my three kids during the day and then teaching Calculus at night. Since the end of the Calculus class in July, I’ve had two weeks where pretty much my only “task” is to hang out with the kids — playing games, doing puzzles, going to the Childrens Museum, etc. It’s been a blessed time, the kind of quality time with one’s kids that a lot of dads only dream about having. But today that period has come to an end with the crossing of a major milestone: The 5-year old, my oldest, just got on the bus for her first day of kindergarten.

Lucy has been intellectually ready for kindergarten for a while now (she went to an excellent Montessori preschool for two years) and has been relishing all summer long the idea that she is heading to kindergarten while her little sister is still in preschool and her brother is still just a baby. So she showed no signs of nervousness, fear, or sadness this morning. As for her dad, though? Not really trepidation, but definitely a sense that both my kid and all of us as a family have crossed over into a major undertaking, namely a minimum thirteen-year journey through the very educational system I have blogged so much about here at this web site. (And if she takes her old man’s route, this becomes a 22-year journey.) And definitely a bit of a lump in my throat as I watched her ride off down the street, and as I sit here now knowing she’s 30 minutes in to her formal education.

It gives me a sudden, deep, and above all deeply personal sense of perspective about things like teacher licensure and school choice and other issues of the K-12 school system we debate about so much. It’s one thing to be a mathematician writing hack jobs articles about K-12 education and quite another to be a dad whose kid is doing the homework, riding the bus, being affected by the decisions of school boards. Perspective doesn’t necessarily make your thinking about these things any more informed, but it does make you think a lot harder about them as the abstractions of issues like licensing, redistricting, and so on become very concrete — as concrete and real as the big yellow bus that pulls up to our next-door neighbors’ house to take my daughter to a school where her lifelong intellectual development is in the balance.

So I hope that, just as my teaching changed for the better (IMO) once I had kids and could see my students as human beings near the end of this educational journey rather than Just Another Freshman Class, I hope that my thinking and writing about schools will change for the better now that my daughter, and by extension both my wife and I, are in it. And to all those parents who are sharing this journey with us — and especially to all the school teachers, administrators, school board members, politicians, etc. whose decisions and actions shape our kids — our family is pulling for you.